Abstract

The phase-conjugation operator is investigated in the context of the canonical operator theory of first-order optics, which was introduced in a previous series of articles. The matrix representation of this operator and its action on the generalized modes are derived, elucidating the relation between forward and reversed propagation in this new context. The resulting formalism makes possible the derivation of the eigenmodes of conventional and phase-conjugated resonators with loss or gain. Various resonator properties—confinement, stability, mode biorthogonality, and real or complex character of the spot-size parameter—simply follow from the guiding-ray-label representation.

© 1983 Optical Society of America

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  1. M. Nazarathy and J. Shamir, "First order optics—a canonical operator representation: lossless systems," J. Opt. Soc. Am. 72, 356–364 (1982).
  2. M. Nazarathy and J. Shamir, "First order optics—operator representation for systems with loss or gain," J. Opt. Soc. Am. 72, 1398–1408 (1982).
  3. M. Nazarathy, A. Hardy, and J. Shamir, "Generalized mode propagation in first-order optical systems with loss or gain," J. Opt. Soc. Am. 72, 1409–1420 (1982).
  4. H. Kogelnik and T. Li, "Laser beams and resonators," Proc. IEEE 54, 1312–1329 (1966).
  5. W. K. Kahn, "Unstable optical resonators," Appl. Opt. 5,407–413 (1966).
  6. A. Yariv and P. Yeh, "Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers," Opt. Commun. 13, 370–374 (1975).
  7. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971).
  8. L. W. Casperson, "Gaussian modes in high loss laser resonators," Appl. Opt. 14, 1193–1199 (1975).
  9. L. W. Casperson, "Synthesis of Gaussian optical beams," Appl. Opt. 20, 2243–2249 (1981).
  10. U. Ganiel and A. Hardy, "Eigenmodes of optical resonators with mirrors having Gaussian reflectivity profiles," Appl. Opt. 15, 2145–2149 (1976).
  11. A. Yariv, Quantum Electronics (Wiley, New York, 1975).
  12. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).
  13. A. E. Siegman, "A canonical formulation for analyzing multielement unstable resonators," IEEE J. Quantum Electron. QE-12, 35–40 (1976).
  14. A. Siegman, "Orthogonality properties of optical resonator eigenmodes," Opt. Commun. 31, 369–373 (1979).
  15. E. M. Wright and W. J. Firth, "Orthogonality properties of general optical resonator eigenmodes," Opt. Commun. 40, 410–412 (1982).
  16. K. E. Oughstun, "On the completeness of the stationary transverse modes in an optical cavity," Opt. Commun. 42,72–76 (1982).
  17. D. Stoler, "Operator-algebraic methods for laser cavity modes," in Coherence and Quantum Optics IV, L. Mandel and E. Wolf, eds. (Addison-Wesley, Reading, Mass., 1976), pp. 369–485.
  18. J. AuYeung, D. Fekete, D. M. Pepper and A. Yariv, "A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors," IEEE J. Quantum Electron. QE-15, 1180–1188 (1980).
  19. J. F. Lam and W. P. Brown, "Optical resonators with phaseconjugate mirrors," Opt. Lett. 5, 61–63 (1980).
  20. P. A. Belanger, A. Hardy, and A. E. Siegman, "Resonant modes of optical cavities with phase conjugate mirrors," Appl. Opt. 19, 602–609 (1980).
  21. P. A. Belanger, A. Hardy, and A. E. Siegman, "Resonant modes of optical cavities with phase conjugate mirrors: higher order modes," Appl. Opt. 19, 479–481 (1980).
  22. A. E. Siegman, P. A. Belanger, and A. Hardy, "Optical resonators using phase conjugate mirrors," in Optical Phase Conjugation, R. H. Fisher, ed. (Academic, New York, 1983).
  23. A. Hardy, P. A. Belanger and A. E. Siegman, "Orthogonality properties of phase conjugate optical resonators," Appl. Opt. 21, 1122–1124 (1982).
  24. W. Shao-Min and H. Weber, "Aspherical resonator equivalent to arbitrary phase-conjugate resonators," Opt. Commun. 41, 360–362 (1982).
  25. M. Nazarathy, J. Shamir, and A. Hardy, "Phase-conjugate-mirror resonators—a canonical operator analysis," J. Opt. Soc. Am. 72, 410(A) (1982).
  26. M. Nazarathy, J. Shamir, and A. Hardy, "Nonideal phase conjugate resonators—a canonical operator analysis," J. Opt. Soc. Am. 73, 587–593 (1983).
  27. M. Nazarathy and J. Shamir, "Holography described by operator algebra," J. Opt. Soc. Am. 71, 529–541 (1981).
  28. W. D. Montgomery, "Unitary operators in the homogenous wavefield," Opt. Lett. 6, 314–315 (1981).
  29. E. Wolf, "Phase conjugacy and symmetries in spatially bandlimited wavefields containing no evanescent components," J. Opt. Soc. Am. 70, 1311–1319 (1980).
  30. S. C. Sheng and A. E. Siegman, Edward L. Gintzon Laboratory, Stanford University, Stanford, Calif. (personal communication with A. Hardy).
  31. Ref. 12, p. 87.
  32. As was explained in III, on projection on the |x〉 basis, the PCO that appears in Eq. (39) cancels the conjugation operation applied upon one of the functions involved in the usual mathematical definition of scalar product, yielding a scalar product of the type ∫ψnψmdx.

1983 (1)

1982 (8)

1981 (3)

1980 (5)

1979 (1)

A. Siegman, "Orthogonality properties of optical resonator eigenmodes," Opt. Commun. 31, 369–373 (1979).

1976 (2)

U. Ganiel and A. Hardy, "Eigenmodes of optical resonators with mirrors having Gaussian reflectivity profiles," Appl. Opt. 15, 2145–2149 (1976).

A. E. Siegman, "A canonical formulation for analyzing multielement unstable resonators," IEEE J. Quantum Electron. QE-12, 35–40 (1976).

1975 (2)

A. Yariv and P. Yeh, "Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers," Opt. Commun. 13, 370–374 (1975).

L. W. Casperson, "Gaussian modes in high loss laser resonators," Appl. Opt. 14, 1193–1199 (1975).

1966 (1)

Arnaud, J. A.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

AuYeung, J.

J. AuYeung, D. Fekete, D. M. Pepper and A. Yariv, "A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors," IEEE J. Quantum Electron. QE-15, 1180–1188 (1980).

Belanger, P. A.

Brown, W. P.

Casperson, L. W.

Fekete, D.

J. AuYeung, D. Fekete, D. M. Pepper and A. Yariv, "A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors," IEEE J. Quantum Electron. QE-15, 1180–1188 (1980).

Firt, W. J.

E. M. Wright and W. J. Firth, "Orthogonality properties of general optical resonator eigenmodes," Opt. Commun. 40, 410–412 (1982).

Ganiel, U.

Hardy, A.

Kahn, W. K.

Kogelnik, H.

H. Kogelnik and T. Li, "Laser beams and resonators," Proc. IEEE 54, 1312–1329 (1966).

Lam, J. F.

Li, T.

H. Kogelnik and T. Li, "Laser beams and resonators," Proc. IEEE 54, 1312–1329 (1966).

Montgomery, W. D.

Nazarathy, M.

Oughstun, K. E.

K. E. Oughstun, "On the completeness of the stationary transverse modes in an optical cavity," Opt. Commun. 42,72–76 (1982).

Pepper, D. M.

J. AuYeung, D. Fekete, D. M. Pepper and A. Yariv, "A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors," IEEE J. Quantum Electron. QE-15, 1180–1188 (1980).

Shamir, J.

Shao-Min, W.

W. Shao-Min and H. Weber, "Aspherical resonator equivalent to arbitrary phase-conjugate resonators," Opt. Commun. 41, 360–362 (1982).

Sheng, S. C.

S. C. Sheng and A. E. Siegman, Edward L. Gintzon Laboratory, Stanford University, Stanford, Calif. (personal communication with A. Hardy).

Siegman, A.

A. Siegman, "Orthogonality properties of optical resonator eigenmodes," Opt. Commun. 31, 369–373 (1979).

Siegman, A. E.

A. Hardy, P. A. Belanger and A. E. Siegman, "Orthogonality properties of phase conjugate optical resonators," Appl. Opt. 21, 1122–1124 (1982).

P. A. Belanger, A. Hardy, and A. E. Siegman, "Resonant modes of optical cavities with phase conjugate mirrors," Appl. Opt. 19, 602–609 (1980).

P. A. Belanger, A. Hardy, and A. E. Siegman, "Resonant modes of optical cavities with phase conjugate mirrors: higher order modes," Appl. Opt. 19, 479–481 (1980).

A. E. Siegman, "A canonical formulation for analyzing multielement unstable resonators," IEEE J. Quantum Electron. QE-12, 35–40 (1976).

A. E. Siegman, P. A. Belanger, and A. Hardy, "Optical resonators using phase conjugate mirrors," in Optical Phase Conjugation, R. H. Fisher, ed. (Academic, New York, 1983).

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971).

S. C. Sheng and A. E. Siegman, Edward L. Gintzon Laboratory, Stanford University, Stanford, Calif. (personal communication with A. Hardy).

Stoler, D.

D. Stoler, "Operator-algebraic methods for laser cavity modes," in Coherence and Quantum Optics IV, L. Mandel and E. Wolf, eds. (Addison-Wesley, Reading, Mass., 1976), pp. 369–485.

Weber, H.

W. Shao-Min and H. Weber, "Aspherical resonator equivalent to arbitrary phase-conjugate resonators," Opt. Commun. 41, 360–362 (1982).

Wolf, E.

Wright, E. M.

E. M. Wright and W. J. Firth, "Orthogonality properties of general optical resonator eigenmodes," Opt. Commun. 40, 410–412 (1982).

Yariv, A.

J. AuYeung, D. Fekete, D. M. Pepper and A. Yariv, "A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors," IEEE J. Quantum Electron. QE-15, 1180–1188 (1980).

A. Yariv and P. Yeh, "Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers," Opt. Commun. 13, 370–374 (1975).

A. Yariv, Quantum Electronics (Wiley, New York, 1975).

Yeh, P.

A. Yariv and P. Yeh, "Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers," Opt. Commun. 13, 370–374 (1975).

Appl. Opt. (7)

IEEE J. Quantum Electron. (2)

J. AuYeung, D. Fekete, D. M. Pepper and A. Yariv, "A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors," IEEE J. Quantum Electron. QE-15, 1180–1188 (1980).

A. E. Siegman, "A canonical formulation for analyzing multielement unstable resonators," IEEE J. Quantum Electron. QE-12, 35–40 (1976).

J. Opt. Soc. Am. (7)

Opt. Commun. (5)

W. Shao-Min and H. Weber, "Aspherical resonator equivalent to arbitrary phase-conjugate resonators," Opt. Commun. 41, 360–362 (1982).

A. Siegman, "Orthogonality properties of optical resonator eigenmodes," Opt. Commun. 31, 369–373 (1979).

E. M. Wright and W. J. Firth, "Orthogonality properties of general optical resonator eigenmodes," Opt. Commun. 40, 410–412 (1982).

K. E. Oughstun, "On the completeness of the stationary transverse modes in an optical cavity," Opt. Commun. 42,72–76 (1982).

A. Yariv and P. Yeh, "Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers," Opt. Commun. 13, 370–374 (1975).

Opt. Lett. (2)

Other (9)

A. E. Siegman, P. A. Belanger, and A. Hardy, "Optical resonators using phase conjugate mirrors," in Optical Phase Conjugation, R. H. Fisher, ed. (Academic, New York, 1983).

S. C. Sheng and A. E. Siegman, Edward L. Gintzon Laboratory, Stanford University, Stanford, Calif. (personal communication with A. Hardy).

Ref. 12, p. 87.

As was explained in III, on projection on the |x〉 basis, the PCO that appears in Eq. (39) cancels the conjugation operation applied upon one of the functions involved in the usual mathematical definition of scalar product, yielding a scalar product of the type ∫ψnψmdx.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971).

D. Stoler, "Operator-algebraic methods for laser cavity modes," in Coherence and Quantum Optics IV, L. Mandel and E. Wolf, eds. (Addison-Wesley, Reading, Mass., 1976), pp. 369–485.

H. Kogelnik and T. Li, "Laser beams and resonators," Proc. IEEE 54, 1312–1329 (1966).

A. Yariv, Quantum Electronics (Wiley, New York, 1975).

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

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